2006 B. Borchert and K. Reinhardt introduced a stronger version of determinism
which they called Sudoku-determinism~\cite{borchert2006deterministically}.
They also considered a more general deterministic process than the one
presented in Section~\ref{deterministicprocess}. 

The Sudoku-determinism is defined in a process that reduces a picture step
by step. The difference to the determinism described in the section above is
that instead of determining the pre-image symbol on a position in one shot, they
keep a set of possible pre-image symbols for each position and reduce one set
per step of the process by excluding impossible pre-images. 

Therefore let $T = (\Sigma, \Gamma, \Theta, \pi)$ be a domino tiling
system. For a picture $p \in \Sigma$, $s_p$ is a local picture of the same size
in which every position $(i, j)$ is initialized by the set $s_p(i, j) :=
\pi^{-1}(p(i, j)) \in 2^\Gamma$ of possible pre-image symbols. With this
information, one step in a Sudoku-deterministic process is defined as follows:
\begin{definition} For $s, s' \in (2^\Gamma)^{*,*}$ with $l_1(s) = l_1(s')$ and
$l_2(s) = l_2(s')$, we allow a step $\hat{s} \underset{sd(T)}{\Rightarrow}
\hat{s}'$ if the following conditions hold for all positions $(i, j)$ of $s'$:
\begin{align*}
s'(i,j) = &\{x \in s(i,j) \mid \exists \gamma_1, \gamma_2, \gamma_3,
\gamma_4 \in \Gamma \cup \{\#\} \text{ such that } \\
&\gamma_1 \in \hat{s}(i + 1, j), \gamma_2 \in \hat{s}(i - 1, j),
\gamma_3 \in \hat{s}(i, j + 1), \gamma_4 \in \hat{s}(i, j - 1 )\\
&\text{ and }
\begin{tabular}{|A{0.38cm}|A{0.38cm}|}
\hline
 x & $\gamma_1$ \tabularnewline
\hline 
\end{tabular}, 
\begin{tabular}{|A{0.38cm}|A{0.38cm}|} 
\hline 
$\gamma_2$ & x \tabularnewline
\hline 
\end{tabular}, 
\begin{tabular}{|A{0.38cm}|} 
\hline 
x \tabularnewline 
\hline 
$\gamma_3$ \tabularnewline
\hline 
\end{tabular}, 
\begin{tabular}{|A{0.38cm}|} 
\hline 
$\gamma_4$ \tabularnewline 
\hline 
x \tabularnewline
\hline 
\end{tabular}
\in \Theta\}.
\end{align*}
\end{definition}
One step in a deterministic process which is described in
Section~\ref{deterministicprocess} can be formulated as a special
case~\cite{borchert2006deterministically}.
\begin{definition}
For $s, s', s'' \in (2^\Gamma)^{*,*}$ with $l_1(s) = l_1(s')$ and
$l_2(s) = l_2(s')$ we allow a step $\hat{s} \underset{d(T)}{\Rightarrow}
\hat{s}'$ if $\hat{s} \underset{sd(T)}{\Rightarrow} \hat{s}''$ and for all
positions $(i, j)$ of $s'$ we define $s'(i, j) := s''(i, j)$ if $\left| s''(i,
j) \right| = 1$ or $\hat{s}'(i, j) = \hat{s}(i, j)$
\end{definition}
In the following we use the notion from~\cite{borchert2006deterministically},
where $\{p\}$ describes a picture which has the same size as $p$ and has a
singleton set $\{p(i, j)\}$ instead of the letter $p(i, j)$ on every position.

A Sudoku-deterministically recognizable picture language $L_{sd}(T)$ over a
domino tiling system $T = (\Sigma, \Gamma, \Theta, \pi)$ is $L_{sd}(T) := \{p
\in \Sigma^{*,*} \mid \exists p' \in L(T)$ s.t. $\hat{s}_p
\overset{*}{\underset{sd(T)}{\Rightarrow}} \{\hat{p}'\}\}$. That means that the
local picture $\hat{s}_p$ can be transformed with a finite number of steps
of a Sudoku-deterministic process into the picture $\{\hat{p}'\}$. The language
$L_{d}(T)$ is defined in the same way using
$\overset{*}{\underset{d(T)}{\Rightarrow}}$ instead of
$\overset{*}{\underset{sd(T)}{\Rightarrow}}$. The family of all
Sudoku-deterministically recognizable picture languages is denoted by SDREC. The
class of all deterministically recognizable picture languages defined in this
section is denoted by DREC. Remark that we have got two classes that are denoted
by DREC.

Let's now regard an example of a DS $T$, some subpictures $t_p, t'_p, t''_p
\in B_{2, 2}(s_p)$ with $p \in \Sigma$ and how a Sudoku-deterministic and a
deterministic process work with this subpictures depending on the given $T$. For
convenience we only test two of the four possible neighbours of every position
in the subpictures.
\begin{example}
For a DS $T = (\Sigma, \Gamma, \Delta, \pi)$ with $\Gamma = \{a, b, c, d\}$ and
$\Delta = \{$
\begin{tabular}{|A{0.38cm}|} 
\hline 
$a$ \tabularnewline 
\hline 
$b$ \tabularnewline
\hline 
\end{tabular}, 
\begin{tabular}{|A{0.38cm}|} 
\hline 
$b$ \tabularnewline 
\hline 
$a$ \tabularnewline
\hline 
\end{tabular},\begin{tabular}{|A{0.38cm}|} 
\hline 
$c$ \tabularnewline 
\hline 
$d$ \tabularnewline
\hline 
\end{tabular},\begin{tabular}{|A{0.38cm}|} 
\hline 
$d$ \tabularnewline 
\hline 
$c$ \tabularnewline
\hline 
\end{tabular}, 
\begin{tabular}{|A{0.38cm}|} 
\hline 
$x$ \tabularnewline 
\hline 
$x$ \tabularnewline
\hline 
\end{tabular},
\begin{tabular}{|A{0.38cm}|} 
\hline 
$x$ \tabularnewline 
\hline 
\# \tabularnewline
\hline 
\end{tabular},
\begin{tabular}{|A{0.38cm}|} 
\hline 
\# \tabularnewline 
\hline 
$x$ \tabularnewline
\hline 
\end{tabular},
\begin{tabular}{|A{0.38cm}|A{0.38cm}|} 
\hline 
$a$ & $c$ \tabularnewline
\hline 
\end{tabular},
\begin{tabular}{|A{0.38cm}|A{0.38cm}|} 
\hline 
$c$ & $c$ \tabularnewline
\hline 
\end{tabular},
\begin{tabular}{|A{0.38cm}|A{0.38cm}|} 
\hline 
$b$ & $d$ \tabularnewline
\hline 
\end{tabular},
\begin{tabular}{|A{0.38cm}|A{0.38cm}|} 
\hline 
$d$ & $b$ \tabularnewline
\hline 
\end{tabular},
\begin{tabular}{|A{0.38cm}|A{0.38cm}|} 
\hline 
$x$ & $x$ \tabularnewline
\hline 
\end{tabular}, 
\begin{tabular}{|A{0.38cm}|A{0.38cm}|} 
\hline 
$x$ & \# \tabularnewline
\hline 
\end{tabular},
\begin{tabular}{|A{0.38cm}|A{0.38cm}|} 
\hline 
\# & $x$ \tabularnewline
\hline 
\end{tabular}
$\mid x \in \Gamma\}$ holds
\begin{center}
$t_p = $ 
\begin{tabular}{|C{1.6cm}|C{1.6cm}|} 
\hline 
$a, b, c, d$ & $a, b, c, d$ \tabularnewline
\hline 
$b, d$ & $a, b, c, d$ \tabularnewline
\hline 
\end{tabular} $\underset{sd(T)}{\Rightarrow}$
\begin{tabular}{|C{1.6cm}|C{1.6cm}|} 
\hline 
$a, b, c, d$ & $a, b, c, d$ \tabularnewline
\hline 
$b, d$ & $b, d$ \tabularnewline
\hline 
\end{tabular}
\end{center}
\begin{center}
$t_p' = $ 
\begin{tabular}{|C{1.6cm}|C{1.6cm}|} 
\hline 
$a, b, c, d$ & $c, d$ \tabularnewline
\hline 
$a, b, c, d$ & $a, b, c, d$ \tabularnewline
\hline 
\end{tabular} $\underset{sd(T)}{\Rightarrow}$
\begin{tabular}{|C{1.6cm}|C{1.6cm}|} 
\hline 
$a, b, c, d$ & $a, b, c, d$ \tabularnewline
\hline 
$a, b, c, d$ & $c, d$ \tabularnewline
\hline 
\end{tabular}
\end{center}
\begin{center}
$t_p'' = $ 
\begin{tabular}{|C{1.6cm}|C{1.6cm}|} 
\hline 
$a, b, c, d$ & $c, d$ \tabularnewline
\hline 
$b, d$ & $a, b, c, d$ \tabularnewline
\hline 
\end{tabular} $\underset{d(T)}{\Rightarrow}$
\begin{tabular}{|C{1.6cm}|C{1.6cm}|} 
\hline 
$a, b, c, d$ & $c, d$ \tabularnewline
\hline 
$b, d$ & $d$ \tabularnewline
\hline 
\end{tabular}
\end{center}
\end{example}
Furthermore, the language family dependencies will be talked about. The
following class relationships are proved
in~\cite{borchert2006deterministically}.
\begin{theorem}
DREC $\subseteq$ SDREC, \\
DREC $\subseteq$ REC, \\
DREC $\subseteq$ co-REC.
\end{theorem}
The first statement is proved by constructing a DS $T' = (\Sigma, \Gamma',
\Delta', \pi')$ to the given DS $T = (\Sigma, \Gamma, \Delta, \pi)$ such that
$L_{sd}(T') = L(T)$. The second statement has been proved by defining
a TS $T$ that $L_{d}(T') = L(T)$ and to prove the third statement it is used a
TS $T'$ similar to $T$ that follows co-DREC $\subseteq$ REC. Remark that
co-DREC$ = \{L \mid \bar{L} \in DREC\}$ and co-REC$ = \{L \mid \bar{L} \in
REC\}$.
In~\cite{prusa2013new} D.
Pr{\r{u}\v{s}}a, F. Mr\'{a}z and F. Otto were designing a 2DSA that simulates a
Sudoku-deterministic process. This implies the following theorem.
\begin{theorem}
SDREC $\subset \familyOf{2DSA}$
\end{theorem}